Recently, Talenti's comparison for Dirichlet boundary was generalized to the compact manifolds with positive Ricci curvature by Colladay, Langford and McDonald [1]}, and to complete noncompact manifold with nonnegative Ricci curvature and positive asymptotic volume ratio by Chen and Li [2]}. Also Talenti's comparison for Robin boundary was proved in both Euclidean space [3]}, [4]} and complete manifolds [5]}, [6]}. To the best of our knowledge, Bandle's comparison, however, did not receive as much attention as Talenti's comparison on manifolds. The purpose of the present paper is to study the Bandle's comparison for solutions to parabolic equation on manifolds. In particular, we will establish a sharp comparison between the solutions to (REF ) and (REF ) on complete and noncompact manifolds with nonnegative Ricci curvature and positive asymptotic volume ratio, and compact manifolds with positive Ricci curvature, see Theorem REF  below.
